Title: Spontaneous Crystal Thermal Hall Effect in Insulating Altermagnets

Abstract: We show that magnetic insulators with a collinear and compensated order can exhibit a thermal Hall effect even at zero magnetic field if they have altermagnetic symmetries. We predict a finite thermal Hall conductivity vector $\boldsymbol{\kappa}_\text{H}$ for a rutile-inspired effective spin model with Dzyaloshinskii-Moriya interaction. Within the linear spin-wave theory, we identify two magnon branches that carry identical Berry curvature and give rise to a finite $\boldsymbol{\kappa}_\text{H}$, which can be controlled by the N\'eel vector orientation and by strain. The thermal Hall response is further complemented with a spin Nernst response to contrast spin and heat transport in altermagnetic insulators with those in ferromagnets and antiferromagnets. Our results establish the crystal thermal Hall effect of magnons and we discuss material candidates for experimental realization, such as MnF$_2$, CoF$_2$, and NiF$_2$.